Papers › Classification of Small Triorthogonal Codes
Classification of Small Triorthogonal Codes
Sepehr Nezami, Jeongwan Haah
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Triorthogonal codes are a class of quantum error correcting codes used in magic state distillation protocols. We classify all triorthogonal codes with n+k ≤38, where n is the number of physical qubits and k is the number of logical qubits of the code. We find $38$ distinguished triorthogonal subspaces and show that every triorthogonal code with n+k≤38 descends from one of these subspaces through elementary operations such as puncturing and deleting qubits. Specifically, we associate each triorthogonal code with a Reed-Muller polynomial of weight n+k, and classify the Reed-Muller polynomials of low weight using the results of Kasami, Tokura, and Azumi and an extensive computerized search. In an appendix independent of the main text, we improve a magic state distillation protocol by reducing the time variance due to stochastic Clifford corrections.
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